It is rather a poor circuit. My dislike stems from the fact that it supplies unregulated voltage to float charge the battery. For a sealed lead-acid (SLA) battery, the float voltage is critical, too high and you will dry out the electrolyte and corrode the lead grid, too low, and the battery will be less than 90% charged causing the plates to sulphate.
A better design would utilize a battery charger chip like the PB137 to regulate the float charge voltage (13.70V for a six-cell battery). For a three-cell battery, the nominal float charge voltage should be 6.85V, and you could use a LM317 with a trim pot to set the float voltage.
I would start with a DC wall wart that puts out a nominal 6 to 9V. Use that to drive the float regulator. Diode-OR the 6V battery voltage with the wall-wart output as an input to a low-dropout regulator which creates the final 5V. Since you don't have much headroom between the battery and the low-dropout regulator, use a Schottky diode in that path.